Theorem 1.1.

The following statements hold for system (1.1).

1. 1.

   If c = a = 0 and b = 1, it is completely integrable with the two independent first integrals H₀ = x² + (z + 1)² and H₁ = x² − y².

2. 2.

   If b = 1, a = 0 and c ≠ 0, it has the first integral H₁ = x² − y².

3. 3.

   If b = 1 and a ≠ 0 it has the invariant I₁ = H₁e^2at = (x² − y²)e^2at.

4. 4.

   If c = −a with a ≠ 0 it has the invariant I₂ = (b(x² − y² − z²) + z²)e^2at.

Theorem 1.1 will be used in the following sections for studying the global dynamics of system (1.1) having a first integral or an invariant. As any polynomial differential system, system (1.1) can be extended to an analytic system on a closed ball of radius one, whose interior is diffeomorphic to ℝ³ and its invariant boundary, a two-dimensional sphere 𝕊² plays the role of the infinity of ℝ³. This ball will be denoted by B and called the Poincaré ball, due to the fact that the technique for doing so is the Poincaré compactification, which is described in detail in [1] (in fact in that paper the compactification is done for ℝⁿ, and Poincaré in [10] did it in ℝ²).

Two polynomial vector fields are said to be topologically equivalent if there exists a homeomorphism on the closed Poincaré ball carrying orbits of the flow induced on the Poincaré ball by the first vector field into orbits of the flow induced in the Poincaré ball by the second vector field preserving or reversing the orientation of all the orbits.

By using this compactification technique for the dynamics of system (1.1), we can prove the next result.

Theorem 1.2.

For all values of the parameters a, b, c ∈ ℝ with b > 0, the phase portrait of system (1.1) on the sphere at infinity, 𝕊², is topologically equivalent to that of Figure 1. In that figure we can see that there exist four centers at the positive and negative endpoints of the x- and y-axis and two hyperbolic saddles at the positive and negative endpoints of the z-axis.

Note that the dynamics at infinity does not depend on the parameter values.

Now we continue the study of the dynamics of system (1.1) for the values of the parameters in Theorem 1.1. that is when b = 1 or a + c = 0 with a ≠ 0 (and b ≠ 1). We provide the description of the global dynamics of this polynomial differential system in its compactification on the Poincaré ball and we describe the α- and ω-limit sets of all orbits of system (1.1) for some values of the parameters.

Theorem 1.3.

The dynamics of the three-dimensional forced-damped differential system (1.1) in the Poincaré ball under the four assumptions given in the statement of Theorem 1.1 are described in section 4.

The paper is organized as follows. In section 2 we summarize several preliminary results of system (1.1). In section 3 we prove Theorem 1.2 by means of the Poincaré compactification technique, and in section 4 we prove Theorem 1.3.

Lemma 2.1.

If ϕ(t) = (x(t),y(t),v(t)) is a trajectory of the compactified system (1.1) with a ≠ 0 and c = −a, or a ≠ 0 and b = 1, then its α-limit set α(ϕ) and its ω-limit set ω(ϕ) are contained in the closure in the Poincaré ball of the sets {H₁ = x² − y² = 0} when b = 1, and {H₂ = b(x² − y² − z²) + z² = 0} when c = −a ≠ 0 (see Theorem 1.1).

Proof.

Assume first that a > 0. Let q₁ ∈ ω(ϕ). Then there exists a sequence {t_n} such that t_n → ∞ and ϕ(t_n) → q₁. Thus, by statements 4 and 5 of Theorem 1.1 we have

Suppose now that q₂ ∈ α(ϕ). Then there exists a sequence {t_n} such that t_n → −∞ and ϕ(t_n) → q₂. Hence

Thus the constant κ is zero and then H_i(ϕ(t_n)) · e^2at_n → 0. This implies that H_i(ϕ(t_n)) → 0 and so q₂ is in the closure in the Poincaré ball of the set {H_i = 0}.

The case a < 0 can be done in the same manner reverting the roles of q₁ and q₂.

Lemma 2.2.

For all values of the parameters a, b, c the z-axis is an invariant set of system (1.1). The flow restricted to this invariant straight line is as follows: if c < 0 the origin (0,0,0) is a global attractor. If c > 0 then the origin is a global repeller. If c = 0 all points in the z-axis are singular.

Proof.

It follows directly from system (1.1).

Now we study the finite singular points whenever a ≠ 0 and c = −a or ac ≠ 0 and b = 1. Let p be a hyperbolic singular point of system (1.1) whose Jacobian matrix has k eigenvalues with negative real part. Then its stability index is k. Of course, 0 ⩽ k ⩽ 3. The topology index of p is given by (−1)^k (see Theorem 8.1 of [5]).

Lemma 2.3.

Consider system (1.1) with b = 1 and ac ≠ 0.

1. 1.

   If c < 0 and a > 1 the unique singular point is the origin which has stability index 3 (an attractor).

2. 2.

   If c < 0 and a = 1 the unique singular point is the origin whose eigenvalues of its Jacobian matrix are −2, 0, c.

3. 3.

   If c < 0 and a ∈ (−1,1) it has three singular points: the origin and R±0=(±c(a−1),±c(a−1),a−1). The origin has stability index one and the points R±0 both have stability index two if a < 0 and stability index three if a > 0.

4. 4.

   If c < 0 and a = −1 it has three singular points: the origin whose eigenvalues of its Jacobian matrix are 2, 0, c, and the two points (±−2c, ±−2c,−2) which both have stability index two.

5. 5.

   If c < 0 and a < −1 it has five singular points: the origin which has stability index 1, R±0 given in statement 3. which have both stability index 1, and R±1=(∓c(a+1),±c(a+1),−a−1) which have both stability index two.

6. 6.

   If c > 0 and a > 1 it has five singular points which are the origin (with stability index two), R±0 and R±1 having the four of them stability index one.

7. 7.

   If c > 0 and a = 1 it has two singular points which are the origin (whose eigenvalues of its Jacobian matrix are −2, 0, c and (±−2c, ∓−2c,−2) which have both stability index one).

8. 8.

   If c > 0 and a ∈ (−1,1) it has three singular points which are the origin and R±1 and all of them have stability index one.

9. 9.

   If c > 0 and a = −1 it has the origin as its unique singular point whose eigenvalues of its Jacobian matrix are 2, 0, c.

10. 10.

    If c > 0 and a < −1 it has the origin as its unique singular point which has stability index zero (a repeller).

Proof.

The proof of the lemma follows by direct calculations.

Lemma 2.4.

Consider system (1.1) with c = −a, a ≠ 0 and b ≠ 1:

1. 1.

   If a > 1 it has the origin as its unique singular point which has stability index three (an attractor).

2. 2.

   If a = 1 it has the origin as its unique singular point whose eigenvalues of its Jacobian matrix are −2, −1, 0.

3. 3.

   If a ∈ (−1,1) it has three singular points which are the origin and

   S±=(±B,±(−1+b−A)B2a,−1−b+A2b)

   with

   A=1+b(b+4a2−2),   B=−1−b(2a2−1)+A2b3.

   All three points have stability index two when a ∈ (0,1) and stability index one when a ∈ (−1,0).

4. 4.

   If a = −1 it has the origin as its unique singular point whose eigenvalues of its Jacobian matrix are 2, 1, 0.

5. 5.

   If a < −1 it has the origin as its unique singular point which has stability index zero (a repeller).

Proof.

The proof of the lemma follows by direct computations.

3.1. Study of the infinity in the local charts U₁ and V₁

The Poincaré compactification p(X) of system (1.1) in the local chart U₁ is given by

Note that the unique singular point of system (3.1) on z₃ = 0 is the origin (0,0,0) and the eigenvalues of the linear part of the system at this point are ±ib, and 0. System (3.1) restricted to z₃ = 0 (because this corresponds to the sphere at infinity) reduces to

Fig. 2.

Phase portrait of system (3.2).

3.2. Study of the infinity in the local charts U₂ and V₂

The expression of the Poincaré compactification in the local chart U₂ of system (1.1) writes as

Taking z₃ = 0, system (3.3) becomes

Again we observe that the flow on the local chart V₂ is the same as the flow on the local chart U₂ reversing the time, because the compactified vector field p(X) in V₂ coincides with the vector field p(X) in U₂ multiplied by −1.

3.3. Study of the infinity in the local charts U₃ and V₃

The expression of the Poincaré compactification of system (1.1) in the local chart U₃ is given by

If z₃ = 0 the unique singular point of system (3.5) is p = (0,0,0). We want to study the local flow of this system around p. The eigenvalues of the linear part at p are ±b and 0. Hence, system (3.5) has a saddle at p when we restrict the system to the infinity (that is to z₃ = 0 which is invariant) and a one-dimensional center manifold at p contained in the interior of the ball diffeomorphic to ℝ³. Note that if we consider z₁ = z₂ = 0 then system (3.5) reduces to z˙₁ = 0 z˙₂ = 0 and z˙3=−cz32. So there exists a trajectory of system (1.1) which escapes to infinity as t → ±∞ (depending on the sign of c). In fact since the infinity of system (1.1) in the local chart U₃ is invariant, the unique way in order that a solution reaches the infinity is tending to the singular point p (a saddle at the sphere at infinity) and this is possible only over the center manifold z₁ = z₂ = 0. In the local chart V₃ we have to change the sign of system (3.5) because the compactified vector field p(X) in V₃ is the same as in U₃ multiplied by −1. So in V₃ the dynamics on the z₃-axis is governed by the equation z˙3=cz32.

Proof.

[Proof of Theorem 1.2] Considering the analysis made in the previous sub-sections we obtain the phase portrait of system (1.1) on the sphere at infinity: the system has four centers, localized at the endpoints of the x and y-axis, and two saddles localized at the endpoints of the z-axis (see Figure 1). Moreover, the orbits of the system may come and go to infinity along a one-dimensional center manifold of these saddle points, depending on the sign of the parameter c. The dynamics near the center at infinity is more complex, due to the periodic orbits.

Fig. 1.

Phase portrait of system (1.1) on the sphere at infinity.

4.1. Case b = 1, a = c = 0

In this case system (1.1) is completely integrable. There are three straight lines filled of singularities (0,0,z), (x,0,−1) and (0,y,−1). The endpoints of the straight line (0,0,z) on the Poincaré sphere are (0,0,±1), the endpoints of the straight line (x,0,1) are (±1,0,0), and the endpoints of the straight line (0,y,−1) are (0,±1,0). According to the introduction the trajectories are in the intersection of the elliptic cylinder x² + (z + 1)² = c₁ with the hyperbolic cylinder x² − y² = c₂, where c₁ ⩾ 0 and c₂ ∈ ℝ. The intersection produces families of periodic orbits, and for c₁ = c₂ the intersection occurs on the straight line (x,0,−1).

4.2. Case b = 1, a = 0, c ≠ 0

In this case H₁ = x² − y² is a first integral. We study the dynamics of system (1.1) restricted to the hyperbolic cylinders x² − y² = ±r² if r ≠ 0, and on the planes (x − y)(x + y) = 0 for r = 0.

For r = 0 the level is formed by two invariant planes with endpoints being the separatrices of the saddles on the Poincaré sphere. In the plane y = −x system (1.1) becomes

If c < 0 the unique singular point of system (4.1) is the origin (0,0) which is a stable node.

Now we study the phase portrait in a neighborhood of the infinity. Since system (4.1) is defined in ℝ² we study it doing the Poincaré compactification in the Poincaré disc, see for more details on the Poincaré disc, Chapter 5 of [3]. System (4.1) in the local chart U₁ given by the Poincaré compactification is

Note that there are no infinite singular points on U₁. On the other hand system (4.1) on the local chart U₂ is

We study the origin of U₂ which is a semi-hyperbolic singular point. Applying Theorem 2.19 of [3] we get that it is a saddle-node with central manifold on z₁ = 0 (where the flow is increasing), and the stable separatrices are located on z₂ = 0. Since system (4.1) has degree two, the local phase portrait on V₂ have the opposite stability.

Since the unique finite singular point of system (4.1) is the origin and the straight line x = 0 is invariant, the system has no limit cycles. In short, by the Poincaré-Bendixson theorem (see for instance Corollary 1.30 of [3]) the unstable separatrix of the saddle-node at the origin of U₂ has ω-limit the finite singular point (0,0). See the phase portrait in the Poincaré disc of Figure 3(a).

Fig. 3.

Global phase portrait of system (4.1) for b = 1 and a = 0. (a): with c < 0, (b): with c > 0.

If c > 0 there are two more singular points beyond the origin. The origin is a saddle in this case and the two new singular points are unstable nodes if c − 8 ⩾ 0, or unstable foci if c − 8 < 0. Proceeding in a similar way to the case c < 0, the unique infinite singular points are the origins of U₂ and V₂, which are saddle-nodes, but now the nodal part of the saddle-node is in U₂, see Figure 3(b). The unstable separatrices of the origin are contained in the straight line x = 0 and they go to the origins of U₂ and V₂. By the Poincaré-Bendixson theorem the stable separatrices at the origin have α-limit at the repeller nodes (or foci) in the finite region. This completes the global phase portrait of system (4.1) for c > 0, see Figure 3(b).

In the plane y = x system (1.1) becomes

Note that system (4.2) is the same as system (4.1) doing the reparameterization of the time t → −t and changing the parameter c by −c. This implies that the global phase portraits of system (4.2) are topologically equivalent to the ones shown in Figure 3.

The straight line (0,0,z) (intersection of the planes y = −x and y = x) is invariant with its points having (0,0,0) as their ω-limit.

For r ≠ 0 we have a pair of hyperbolic cylinders. System (1.1) restricted to the hyperbolic cylinder x² − y² = r², r ≠ 0, (taking x=±r2+y2) is, respectively

Note that system (4.3) and system (4.4) are the same doing the change c → −c and reparameterizing the time by t → −t. Then it is sufficient to analyse the phase portraits of system (4.3).

System (4.3) has a unique finite singular point which is a stable node or stable focus if c < 0, and an unstable node or unstable focus if c > 0.

There are no infinite singular points in the local chart U₁. In the local chart U₂ system (4.3) becomes

The origin of system (4.5) is an infinite singular point whose eigenvalues are 0 and 1. Since it is semi-hyperbolic it can be a saddle, a node or a saddle-node. Considering that the sum of the indices of all singular points on the Poincaré sphere is 2 by the Poincaré-Hopf Theorem (see Theorem 6.30 of [3]), and since the finite singular points are two nodes (or two foci), it holds that the origin in U₂ must be a saddle-node. From (4.5) we have z2′|z1=0=−cz22 and then the saddle-node at the origin of U₂ has its unstable separatrix on the z₁-axis. Furthermore for c > 0 the flow is increasing on the z₂-axis and for c < 0 it is decreasing on the z₂-axis.

Doing the change of time dτ=r2+y2dt system (4.3) becomes

Since the divergence of this system is c/r2+y2≠0, system (4.3) has no limit cycles due to the Bendixson criterion, see for instance Theorem 7.1. of [3].

Note that by the Poincaré-Bendixson theorem the unstable separatrix coming from the origin of U₂ must go to the finite attractor. In summary, the global phase portrait for c > 0 (for c < 0 is the same reversing the direction of the orbits) is the one of Figure 4.

Fig. 4.

Global phase portrait of system (4.3).

Furthermore system (1.1) restricted to the hyperbolic cylinder x² − y² = −r², r ≠ 0, (taking y=±r2+x2) is, respectively

Systems (4.6) and (4.7) coincide with systems (4.3) and (4.4), respectively, doing the change of variables (y,z) → (x,z). Then their phase portraits are the same than the one of Figure 4.

The phase portraits in the Poincaré ball for this case can be obtained using the previous information together with Figure 1.

4.3. Case b = 1, a ≠ 0

The planes x = ±y are invariant by the flow of system (1.1). The boundary at infinity of the invariant planes y = ±x coincide exactly with the heteroclinic cycles of the saddles at infinity.

Taking y = x system (1.1) becomes

The infinite singular points of system (4.8) are the same than the infinite singular points of system (4.1).

If c = 0, system (4.8) has the z-axis filled of singular points. The linear part of the system at the singular points (0,z) has the trivial eigenvalues 0 and z + 1 − a. Now by a simple integration of the orbits we get that the orbits of system (4.8) are contained in the ellipses given by the equation

See the phase portraits in the Poincaré disc of system (4.8) with c = 0 in Figure 5(a).

Fig. 5.

Global phase portrait of system (4.8) with a ≠ 0. (a): for c = 0, (b): for c > 0, (a−1) ⩽ 0, (c): forc < 0, (a−1) ⩾ 0, (d): for c > 0, (a − 1) > 0, (e): for c < 0, (a − 1) < 0.

If c ≠ 0 and c(a−1) < 0 the unique singular point of system (4.8) is the origin (0,0). If a−1 > 0 it is a stable node, and if a − 1 < 0 it is an unstable node, see the phase portraits in Figures 5(b) and 5(c).

If c ≠ 0 and a = 1 the unique singular point is the origin which is a semi-hyperbolic node (stable if c < 0 and unstable if c > 0), see the phase portraits in Figures 5(b) and 5(c).

If c ≠ 0 and c(a − 1) > 0 there are two more singular points beyond the origin. The origin is a saddle and the two new singular points are nodes if c(c + 8(1 − a)) ⩾ 0 (stable if c < 0 and unstable if c > 0), or two foci if c(c + 8(1 − a)) < 0 (stable if c < 0 and unstable if c > 0), see the phase portraits in Figures 5(d) and 5(e).

Now we perform the analysis on y = −x. System (1.1) becomes

System (4.9) coincides with system (4.8) doing a reparametrization of the time t → −t and changing the parameters (a,c) → (−a,−c). Then the phase portraits of system (4.9) are topologically equivalent to the ones shown in Figure 5.

Note that in view of Lemma 2.3 system (1.1) under the assumptions of this case can have either two more singular points beyond the origin or four more singular points beyond the origin, two of them located in the plane x = y and the other two on the plane x = −y which are either foci or nodes. Moreover, according to Lemma 2.1 the ω- and α-limit sets of any point not belonging to the planes y = ±x are contained in the closure of these planes inside the Poincaré ball.

4.4. Case c = −a, a ≠ 0, b ≠ 1

We will distinguish between the cases b > 1 and 0 < b < 1 (recall that the case b = 1 is been considered in subsection 4.3 and that b > 0). We consider two cases.

Assume first that b > 1. In this case the cone x² − y² − (b − 1)z²/b = 0 is an invariant algebraic surface. System (1.1) restricted to this cone becomes either

Note that system (4.11) is the same as system (4.10) with a reparameterization of the time t ↦ −t and the parameter a interchanged by −a. So, the behavior is the same with the stability interchanged and the values of a also changed to −a. Hence, we will study only system (4.10).

If |a| ⩾ 1 the unique singular point of system (4.10) is the origin. If a ∈ (−1,1) among the origin there is one more singular point which is

The eigenvalues of the Jacobian matrix at S are

We note that α₁ = 0 if and only if a = 0 which is not possible. Hence, if α₂ ⩾ 0 it is a node (stable if a > 0 and unstable if a < 0), and if α₂ < 0 it is a focus (stable if a > 0 and unstable if a < 0).

Now we study the local phase portraits at the origin of system (4.10). To do so we consider polar coordinates y = r cosθ, z = r sinθ and system (4.10) becomes

Clearly the origin corresponds to r = 0 which is invariant under the flow of system (4.12). On r = 0, taking into account that b > 1, we have two singular points which are θ = 0 and θ = π. Computing the eigenvalues of the Jacobian matrix at these points we get that (0,0) is a saddle if a < 1 and a stable node if a ⩾ 1 (when a = 1 this node is semi-hyperbolic). On the other hand the singular point (0,π) is a saddle if a > −1 and an unstable node if a ⩽ −1 (when a = −1 the node is semi-hyperbolic). To obtain the behavior of the origin of system (4.12) we identify θ = 0 and θ = 2π and r = 0 becomes a circle in the cylinder (r,θ) ∈ ℝ × 𝕊¹. In this case the semi-cylinder r > 0 corresponds to the plane (x,y) of system (4.12) around the origin. So we are only interested in the separatrices (or dynamics) of system (4.12) in r > 0. Going back through the change of variables to obtain the local behavior at the origin, we need to shrink the circle to a point. This will be done for the different values of a. More precisely, when a ⩽ −1 we have the phase portrait in Figure 6(a) with r = 0 as a circle. Shrinking the circle to a point we conclude that the origin is an unstable node, see Figure 6(b). When a ∈ (−1,0)∪(0,1) we have the phase portrait in Figure 7(a) again with r = 0 as a circle. Going back through the change of variables we get that the origin is formed by two hyperbolic sectors, see Figure 7(b). Finally, when a ⩾ 1 we have the phase portrait in Figure 8(a) with r = 0 as a circle. Shrinking the circle to a point we conclude that the origin is a stable node, see Figure 8(b). This concludes the study of the finite singular points of system (4.10).

Fig. 6.

Here b > 1 and a ⩽ −1 (a): Local phase portrait at r = 0 of system (4.12), (b): Local phase portrait at the origin of system (4.10).

Fig. 7.

Here b > 1 and a ∈ (−1,0) ∪ (0,1). (a): Local phase portrait at r = 0 of system (4.12), (b): Local phase portrait at the origin of system (4.10).

Fig. 8.

Here b > 1 and a ⩾ 1. (a): Local phase portrait at r = 0 of system (4.12), (b): Local phase portrait at the origin of system (4.10).

In the local chart U₁ system (4.10) writes as

Since b > 1 system (4.13) has no infinite singular points. On the other hand system (4.10) in the local chart U₂ becomes

Note that the origin of U₂ is not a singular point. So the circle at infinity is a periodic solution.

Doing the rescaling of the time dτ=y2+(b−1)z2/bdt, system (4.10) becomes

Since the divergence of this system is −a/y2+(b−1)z2/b≠0, system (4.10) has no limit cycles. By the Poincaré-Bendixson theorem we get that their phase portraits in the Poincaré disc are given in Figures 9(a) and 9(b).

Fig. 9.

Global phase portraits of system (4.10) with b > 1 and a ≠ 0. (a): for |a| ⩾ 1, (b): a ∈ (−1,0) ∪ (0,1).

According to Lemma 2.1 for any trajectory γ not starting on the cone we have that ω(γ) and α(γ) are contained in the closure of the cone inside the Poincaré ball.

Assume now that 0 < b < 1. Again the cone x² − y² + (1 − b)z²/b = 0 is an invariant algebraic surface. System (1.1) restricted to this cone becomes either

Note that system (4.16) is the same as system (4.15) with a reparameterization of the time t ↦ −t and the parameter a interchanged by −a. So the behavior is the same with the stability interchanged and the values of a also changed to −a. Hence, we will study only system (4.15). However, the study of system (4.15) can be done in an analogous way to system (4.10), obtaining the phase portraits of Figure 9. Again, according to Lemma 2.1 for any trajectory γ not starting on the cone we have that ω(γ) and α(γ) are contained in the closure of the cone inside the Poincaré ball.